Finding Zeros Of Cubic Function G(x)=x^3-7x^2+24x-18 In Exact Form

Jul 12, 2025 by ADMIN 67 views

Finding the Zeros of a Cubic Function in Exact Form

In the realm of mathematics, particularly in algebra, finding the zeros of a polynomial function is a fundamental task. The zeros, also known as roots or x-intercepts, are the values of x for which the function g(x) equals zero. These values provide crucial information about the behavior of the function and its graph. In this article, we will delve into the process of finding the zeros of a specific cubic function, expressing the answers in exact form.

We are given the cubic function:

$g(x) = x^3 - 7x^2 + 24x - 18$

Our goal is to determine all the zeros of this function. Since it's a cubic function, we expect to find up to three zeros, which may be real or complex numbers. We will explore different techniques to solve this problem and express the solutions in their exact form, avoiding decimal approximations.

Methods for Finding Zeros

There are several methods to find the zeros of a polynomial function, and the choice of method often depends on the specific polynomial. For cubic functions, some common approaches include:

Rational Root Theorem: This theorem helps identify potential rational roots (zeros that can be expressed as fractions). It states that if a polynomial has integer coefficients, any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Synthetic Division: Once a potential rational root is identified, synthetic division can be used to test it and, if it is a root, to reduce the polynomial to a lower degree.
Factoring: If we can factor the polynomial, we can easily find the zeros by setting each factor equal to zero.
Cubic Formula: Similar to the quadratic formula, there's a cubic formula that can be used to find the roots of any cubic equation. However, it is quite complex and often less practical for manual calculation.
Numerical Methods: When exact solutions are difficult to find, numerical methods like the Newton-Raphson method can provide approximations of the roots.

Applying the Rational Root Theorem

For our given function, $g(x) = x^3 - 7x^2 + 24x - 18$ , the leading coefficient is 1, and the constant term is -18. According to the Rational Root Theorem, any rational root must be a factor of -18. The factors of -18 are ±1, ±2, ±3, ±6, ±9, and ±18. We can test these potential roots using synthetic division or direct substitution.

Let's start by testing x = 1:

$g(1) = (1)^3 - 7(1)^2 + 24(1) - 18 = 1 - 7 + 24 - 18 = 0$

So, x = 1 is a root of the function. This means that (x - 1) is a factor of g(x).

Using Synthetic Division

Now, we use synthetic division to divide $x^3 - 7x^2 + 24x - 18$ by (x - 1):

1 | 1  -7  24  -18
  |    1  -6   18
  ----------------
    1  -6  18    0

The result of the synthetic division gives us the quotient $x^2 - 6x + 18$ . So, we can write:

$g(x) = (x - 1)(x^2 - 6x + 18)$

Solving the Quadratic Factor

To find the remaining zeros, we need to solve the quadratic equation $x^2 - 6x + 18 = 0$ . We can use the quadratic formula:

$x = rac{-b ± \sqrt{b^2 - 4ac}}{2a}$

Where a = 1, b = -6, and c = 18. Plugging these values into the formula, we get:

$x = rac{-(-6) ± \sqrt{(-6)^2 - 4(1)(18)}}{2(1)}$ $x = rac{6 ± \sqrt{36 - 72}}{2}$ $x = rac{6 ± \sqrt{-36}}{2}$ $x = rac{6 \pm 6i}{2}$ $x = 3 ± 3i$

Thus, the quadratic factor has two complex roots: 3 + 3i and 3 - 3i.

Exact Form of the Zeros

We have found one real root, x = 1, and two complex roots, x = 3 + 3i and x = 3 - 3i. These are the three zeros of the cubic function $g(x) = x^3 - 7x^2 + 24x - 18$ . The zeros are expressed in exact form, meaning we haven't used any approximations.

Conclusion

The zeros of the cubic function $g(x) = x^3 - 7x^2 + 24x - 18$ are 1, 3 + 3i, and 3 - 3i. We found these zeros by applying the Rational Root Theorem, using synthetic division to factor the polynomial, and then solving the resulting quadratic equation. The solutions are expressed in exact form, providing a precise representation of the function's roots. Understanding how to find the zeros of polynomial functions is crucial in various areas of mathematics and its applications, offering insights into the behavior and characteristics of these functions.

The primary objective is to find all the zeros of the given cubic function, which is expressed as $g(x) = x^3 - 7x^2 + 24x - 18$ . The zeros of a function are the values of x for which $g(x) = 0$ . These zeros are crucial as they indicate where the function intersects the x-axis on a graph, and they are fundamental in understanding the function's behavior. In solving this, we will ensure the answers are in exact form, meaning we avoid decimal approximations and keep the roots expressed with radicals or complex numbers where necessary. This process involves several algebraic techniques, including the Rational Root Theorem, synthetic division, and the quadratic formula. Each step is vital in accurately determining the zeros and presenting them in their most precise form.

Initial Steps: The Rational Root Theorem

To kickstart our search for zeros, we employ the Rational Root Theorem. This theorem is a powerful tool that helps us identify potential rational roots of the polynomial. A rational root is a root that can be expressed as a simple fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case, the constant term of $g(x)$ is -18, and the leading coefficient is 1. Thus, the possible rational roots are the factors of -18, which include ±1, ±2, ±3, ±6, ±9, and ±18. These values are our candidates for the roots, and we will test them to see if they make the function equal to zero. Testing these potential roots is a systematic way to narrow down the possible zeros and streamline the solution process. By focusing on these rational possibilities, we can efficiently find the integer and fractional roots of the cubic function.

Testing Potential Roots: Synthetic Division and Evaluation

After identifying potential rational roots, the next step is to test them. A straightforward method for this is direct substitution, where we plug each potential root into the function and see if the result is zero. However, a more efficient technique is synthetic division. Synthetic division not only tells us if a number is a root but also gives us the quotient when the polynomial is divided by (x - root). This is particularly useful for reducing the cubic polynomial to a quadratic, which is easier to solve. Let's start by testing x = 1. Substituting x = 1 into $g(x)$ yields $g(1) = (1)^3 - 7(1)^2 + 24(1) - 18 = 1 - 7 + 24 - 18 = 0$ . This confirms that x = 1 is a root. Now we can use synthetic division to divide $g(x)$ by (x - 1) to find the remaining quadratic factor. This process of testing and dividing allows us to break down the cubic function systematically, making it easier to find all the zeros. Synthetic division is a cornerstone technique in polynomial algebra, enabling efficient root finding and factorization.

Reduction to Quadratic: Performing Synthetic Division

To further simplify the problem, we use synthetic division to divide the original cubic polynomial by the factor we identified, (x - 1). This process allows us to reduce the cubic polynomial to a quadratic polynomial, which is much easier to handle. Setting up synthetic division with 1 as the divisor and the coefficients of $g(x)$ as the dividend, we proceed as follows:

1 | 1  -7  24  -18
  |    1  -6   18
  ----------------
    1  -6  18    0

The numbers in the bottom row represent the coefficients of the quotient and the remainder. The remainder is 0, which confirms that 1 is a root. The quotient is $x^2 - 6x + 18$ . Thus, we can rewrite $g(x)$ as $(x - 1)(x^2 - 6x + 18)$ . Now, we have reduced the problem to finding the zeros of the quadratic factor, which we can solve using the quadratic formula. This reduction is a critical step in solving higher-degree polynomials, making it more manageable by breaking it into simpler parts. Synthetic division serves as an efficient way to factor out known roots and simplify the polynomial equation.

Solving the Quadratic: Applying the Quadratic Formula

With the cubic function reduced to a product of a linear factor and a quadratic factor, the next step is to find the zeros of the quadratic factor, $x^2 - 6x + 18$ . To do this, we employ the quadratic formula, a fundamental tool for solving any quadratic equation of the form $ax^2 + bx + c = 0$ . The quadratic formula is given by:

$x = rac{-b ± \sqrt{b^2 - 4ac}}{2a}$

In our quadratic equation, a = 1, b = -6, and c = 18. Substituting these values into the formula, we get:

$x = rac{-(-6) ± \sqrt{(-6)^2 - 4(1)(18)}}{2(1)}$ $x = rac{6 ± \sqrt{36 - 72}}{2}$ $x = rac{6 ± \sqrt{-36}}{2}$

This simplifies to complex solutions, indicating that the quadratic factor has no real roots. The quadratic formula is indispensable in mathematics, allowing us to find exact solutions even when the roots are not rational or real. It is a cornerstone of algebra and essential for solving polynomial equations.

Simplifying Complex Roots: Expressing in Exact Form

Having applied the quadratic formula, we now simplify the complex roots. From the previous step, we have:

$x = rac{6 ± \sqrt{-36}}{2}$

The square root of -36 can be expressed as $6i$ , where i is the imaginary unit ( $i = \sqrt{-1}$ ). Thus, the equation becomes:

$x = rac{6 \pm 6i}{2}$

Dividing both terms in the numerator by 2, we get:

$x = 3 ± 3i$

Therefore, the complex roots are $3 + 3i$ and $3 - 3i$ . These are the two complex zeros of the cubic function. It’s crucial to express these roots in their exact form, which means keeping them in terms of the imaginary unit i rather than converting them to decimal approximations. This ensures precision and maintains the mathematical integrity of the solutions. Complex roots are a fundamental concept in algebra, arising when the discriminant of a quadratic equation is negative, and they complete the set of zeros for polynomial functions.

Final Answer: Listing All Zeros in Exact Form

Finally, we gather all the zeros of the cubic function $g(x) = x^3 - 7x^2 + 24x - 18$ . We found one real root by testing potential rational roots and then used synthetic division to simplify the polynomial. The real root is x = 1. The remaining quadratic factor was solved using the quadratic formula, yielding two complex roots. These complex roots are x = $3 + 3i$ and x = $3 - 3i$ . Thus, the complete set of zeros for the function, in exact form, is 1, $3 + 3i$ , and $3 - 3i$ . This completes our solution, providing all values of x for which $g(x) = 0$ . Presenting the zeros in exact form is vital for mathematical accuracy and reflects a deep understanding of polynomial behavior and the nature of its roots. In summary, finding zeros involves a combination of techniques, from the Rational Root Theorem to the quadratic formula, each playing a critical role in solving polynomial equations.

Therefore, the zeros of $g(x)$ : 1, 3 + 3i, 3 - 3i.